We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of γ-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.

1. Introduction

Let C be a subset of a real Hilbert space H. Let A:C→H be a nonlinear mapping. The variational inequality problem for A and C is to
(1)findx*∈Csuchthat〈Ax*,v-x*〉≥0,∀v∈C.

The set of solutions of variational inequality problem is denoted by VI⁡(C,A); that is,
(2)VI⁡(C,A)={x*∈C:〈Ax*,x-x*〉≥0,∀x∈C}.

It is well known that variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in variational inequalities, minimax problems, optimization, physics, and the Nash equilibrium problems in noncooperative games. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see, for instance, [1–5] and the references therein.

A mapping A:C⊆H→H is said to be γ-inverse strongly accretive (or γ-inverse strongly monotone) if there exists a positive real number γ such that
(3)〈x-y,Ax-Ay〉≥γ∥Ax-Ay∥2,∀x,y∈C.

If A is γ-inverse strongly accretive, then inequality (3) implies that A is Lipschitzian with constant L:=1/γ; that is, ∥Ax-Ay∥≤(1/γ)∥x-y∥, for all x,y∈C. If in (3) we have that γ=0, then A is called accretive (or monotone).

Let C be a closed and convex subset of a real Hilbert space H. A mapping T:C→H is called a contraction mapping if there exists L∈[0,1) such that ∥Tx-Ty∥≤L∥x-y∥ for all x,y∈C. If L=1, then T is called nonexpansive. A mapping T:C→E is called λ-strictly pseudocontractive of Browder-Petryshyn type [6] if and only if there exists λ∈(0,1) such that
(4)∥Tx-Ty∥2≤∥x-y∥2+λ∥(I-T)x-(I-T)y∥2hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x,y∈C.Tis called pseudocontractive if
(5)∥Tx-Ty∥2≤∥x-y∥2+∥(I-T)x-(I-T)y∥2,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x,y∈C.

We note that inequalities (4) and (5) can be equivalently written as
(6)〈Tx-Ty,x-y〉≤∥x-y∥2-k∥(x-Tx)-(y-Ty)∥2hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀x,y∈C,
for some k>0 and
(7)〈Tx-Ty,x-y〉≤∥x-y∥2∀x,y∈C,
respectively. We remark that T is pseudocontractive if and only if A:=(I-T) is accretive. A point x∈C is a fixed point of T if Tx=x and we denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}.

We observe that in a real Hilbert space H a class of pseudocontractive mappings includes the class of λ-strictly pseudocontractive mappings and hence the classes of nonexpansive and contraction mappings.

Closely related to the variational inequality problems is the problem of finding fixed points of nonexpansive mappings, λ-strict pseudocontraction mappings or pseudocontractive mappings which is the current interest in functional analysis. Several researchers considered a unified approach that approximates a common point of fixed point of nonlinear problems and solutions of variational inequality problems and solutions of variational inequality problems; see, for example, [7–18] and the references therein.

In [19], Takahashi and Toyoda studied the problem of finding a common point of fixed points of a nonexpansive mapping and solutions of a variational inequality problem (1) by considering the following iterative algorithm:
(8)x0∈C,xn+1=αnxn+(1-αn)TPC(xn-λnAxn),n=0,1,…,
where {αn} is a sequence in (0,1), {λn} is a positive sequence, T:C→C is a nonexpansive mapping, and A:C→H is an γ-inverse strongly accretive mapping. They showed that the sequence {xn} generated by (8) converges weakly to some z∈VI⁡(C,A)∩F(S) provided that the control sequences satisfy some restrictions.

Iiduka and Takahashi [20] reconsidered the common element problem via the following iterative algorithm:
(9)x1=x∈C,xn+1=αnx+(1-αn)TPC(xn-λnAxn),n=0,1,…,
where T:C→C is a nonexpansive mapping, A:C→H is a γ-inverse-strongly accretive mapping, {αn} is a sequence in (0,1), and {λn} is a sequence in (0,2α). They proved that the sequence {xn} strongly converges to some point z∈F(T)∩VI⁡(C,A).

Recently, Zegeye and Shahzad [21] investigated the problem of finding a common point of fixed points of a Lipschitz pseudocontractive mapping T and solutions of a variational inequality problem for γ-inverse strongly accretive mapping A by considering the following iterative algorithm:
(10)yn=(1-βn)xn+βnTxn,xn+1=PC[(1-αn)(δnTyn+θnxn+γnPC[I-γA]xn)],
where PC is a metric projection from H onto C and {δn},{θn},{γn},{αn},{βn} are in (0,1) satisfying certain conditions. Then, they proved that the sequence {xn} converges strongly to the minimum-norm point of F(T)∩VI⁡(C,A).

A natural question arises whether we can obtain an iterative scheme which converges strongly to a common point of fixed points of a finite family of pseudocontractive mappings and solutions of a finite family of variational inequality problems for γ-inverse strongly accretive mappings or not.

It is our purpose in this paper to introduce an algorithm and prove that the algorithm converges strongly to a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of variational inequality problems for γ-inverse strongly accretive mappings. The results obtained in this paper improve and extend the results of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21], Theorem 3.2 of Yao et al. [22], and some other results in this direction.

2. Preliminaries

In what follows we will make use of the following lemmas.

Lemma 1.

Letting H be a real Hilbert space, the following identity holds:
(11)∥x+y∥2≤∥x∥2+2〈y,x+y〉,∀x,y∈H.

Lemma 2 (see [23]).

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A:C→E be a γ-inverse strongly accretive mapping. Then, for 0<μ<2γ, the mapping Aμx:=(x-μAx) is nonexpansive.

Lemma 3 (see [24]).

Let C be a nonempty, closed, and convex subset of a smooth Banach space E. Let QC be a sunny nonexpansive retraction from E onto C and let A be an accretive operator of C into E. Then for all λ>0,
(12)VI(C,A)=F(QC(I-λA)).

Lemma 4 (see [25]).

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→E, i=1,…,N, be nonexpansive mappings such that ∩i=1NF(Ti)≠∅. Let T:=θ1T1+θ2T2+⋯+θNTN with θ1+θ2+⋯+θN=1. Then T is nonexpansive and F(T)=∩i=1NF(Ti).

Lemma 5 (see [26]).

Let C be a convex subset of a real Hilbert space H. Let x∈H. Then x0=PCx if and only if
(13)〈z-x0,x-x0〉≤0,∀z∈C.

Lemma 6 (see [27]).

Let C be a closed convex subset of a real Hilbert space H and A:C→C be a continuous pseudo-contractive mapping. Then, for 0<μ<2γ, the mapping Aμx:=(x-μAx) is nonexpansive(i)

F(T) is a closed convex subset of C;

(ii)

(I-T) is demiclosed at zero; that is, if {xn} is a sequence in C such that xn⇀x and Txn-xn→0, as n→∞, then x=T(x).

Lemma 7 (see [28]).

Let H be a real Hilbert space. Then for all xi∈H and αi∈[0,1] for i=1,2,3 such that α1+α2+α3=1 the following equality holds:
(14)∥α1x1+α2x2+α3x3∥2=∑i=13‍αi∥xi∥2-∑1≤i,j≤3‍αiαj∥xi-xj∥2.

Lemma 8 (see [29]).

Let {an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then there exists an increasing sequence {mk}⊂N such that mk→∞ and the following properties are satisfied by all (sufficiently large) numbers k∈N:
(15)amk≤amk+1,ak≤amk+1.

In fact, mk is the largest number n in the set {1,2,…,k} such that the condition an≤an+1 holds.

Lemma 9 (see [30]).

Let {an} be a sequence of nonnegative real numbers satisfying the following relation:
(16)an+1≤(1-αn)an+αnδn,n≥n0,
where {αn}⊂(0,1) and {δn}⊂R satisfying the following conditions: lim⁡n→∞αn=0, ∑n=1∞αn=∞ and limsup⁡n→∞δn≤0. Then, lim⁡n→∞an=0.

3. Main Result

For the rest of this paper, let {an},{bn},{cn},⊂(c,1)⊂(0,1), for some c∈(0,1), and {αn}⊂(0,b)⊂(0,1), for some b∈(0,1), satisfy (i) an+bn+cn=1; (ii) lim⁡n→∞αn=0; and (iii) ∑αn=∞.

Theorem 10.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Tj:C→C, j=1,2,…,M, be Lipschitz pseudocontractive mappings with Lipschitz constants Li, respectively. Let Aj:C→H, for j=1,2,…,N, be γj-inverse strongly accretive mappings. Let f:C→C be a contraction with constant α. Assume that F=[∩j=1MF(Tj)]⋂[∩j=1NVI(C,Aj)] is nonempty. Let a sequence {xn} be generated from an arbitrary x0∈C by
(17)yn=(1-λn)xn+λnTnxn;xn+1=αnf(xn)+(1-αn)(anxn+bnTnyn+cnGxn),
where Tn=Tn(modM) and G:=e0I+e1PC[I-γA1]+e2PC[I-γA2]+⋯+eNPC[I-γAr], for γ∈(0,2γ0), for γ0:=min⁡1≤j≤N{γj} with e0+e1+⋯+er=1 and bn+cn≤λn≤λ<1/(1+L2+1),∀n≥0, for L=max⁡{Lj:1≤j≤M}. Then, {xn} converges strongly to a point x*∈F which is the unique solution of the variational inequality 〈(I-f)(x*),x-x*〉≥0 for all x∈F.

Proof.

From Lemmas 2, 4, and 3 we get that G is nonexpansive mapping with F(G)=∩j=1NVI⁡(C,Aj). Let p∈F. Then from (17), (5), and Lemma 7 we have that
(18)∥yn-p∥2=∥(1-λn)(xn-p)+λn(Tnxn-p)∥2=(1-λn)∥xn-p∥2+λn∥Tnxn-p∥2-λn(1-λn)∥xn-Tnxn∥2≤(1-λn)∥xn-p∥2+λn[∥xn-p∥2+∥xn-Tnxn∥2]-λn(1-λn)∥xn-Tnxn∥2=∥xn-p∥2+λn2∥xn-Tnxn∥2,(19)∥xn+1-p∥2=∥αnf(xn)+(1-αn)(anxn+bnTnyn+cnGxn)-p∥2=∥αn(f(xn)-p)+(1-αn)×(an(xn-p)+bn(Tnyn-p)+cn(Gxn-p))∥2≤αn∥f(xn)-p∥2+(1-αn)×∥an(xn-p)+bn(Tnyn-p)+cn(Gxn-p)∥2≤αn∥f(xn)-p∥2+(1-αn)×[an∥xn-p∥2+bn∥Tnyn-p∥2+cn∥Gxn-p∥2]-(1-αn)bnan∥Tnyn-xn∥2≤αn∥f(xn)-p∥2+(1-αn)[(an+cn)∥xn-p∥2hhhhhhhhhhhhhhhhhhhhhhhhhh+bn∥Tnyn-p∥2]-(1-αn)bnan∥Tnyn-xn∥2≤αn∥f(xn)-p∥2+(1-αn)[(an+cn)∥xn-p∥2hhhhhhhhhhhhh+bn(∥yn-p∥2+∥yn-Tnyn∥2)]-(1-αn)bnan∥Tnyn-xn∥2.

Now, substituting (18) in (19) we get that
(20)∥xn+1-p∥2≤αn∥f(xn)-p∥2+(1-αn)[(an+cn)∥xn-p∥2hhhhhhhhhhhh+bn(∥xn-p∥2+λn2∥xn-Tnxn∥2)hhhhhhhhhhhh+bn∥yn-Tnyn∥2]-(1-αn)bnan∥Tnyn-xn∥2=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2+(1-αn)λn2bn∥xn-Tnxn∥2+(1-αn)bn∥yn-Tnyn∥2-(1-αn)bnan∥Tnyn-xn∥2.

Moreover, from (17), Lemma 7, and Lipschitz property of Tn we get that
(21)∥yn-Tnyn∥2=∥(1-λn)(xn-Tnyn)+λn(Tnxn-Tnyn)∥2=(1-λn)∥xn-Tnyn∥2+λn∥Tnxn-Tnyn∥2-λn(1-λn)∥xn-Tnxn∥2≤(1-λn)∥xn-Tnyn∥2+λnL2∥xn-yn∥2-λn(1-λn)∥xn-Tnxn∥2=(1-λn)∥xn-Tnyn∥2+λn3L2∥xn-Tnxn∥2-λn(1-λn)∥xn-Tnxn∥2=(1-λn)∥xn-Tnyn∥2-λn(1-L2λn2-λn)∥xn-Tnxn∥2.

Substituting (21) into (20) we obtain that
(22)∥xn+1-p∥2≤αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2+(1-αn)bnλn2∥xn-Tnxn∥2+(1-αn)bn[(1-λn)∥xn-Tnyn∥2-λn(1-L2λn2-λn)∥xn-Tnxn∥2]-(1-αn)bnan∥Tnyn-xn∥2,=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-(1-αn)λnbn[1-L2λn2-2λn]∥xn-Tnxn∥2+(1-αn)bn[(1-an)-λn]∥Tnyn-xn∥2=αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2-(1-αn)λnbn[1-L2λn2-2λn]∥xn-Tnxn∥2+(1-αn)bn[bn+cn-λn]∥Tnyn-xn∥2.

But, from the hypothesis we have that
(23)1-2λn-L2λn2≥1-2λ-L2λ2>0,bn+cn≤λn,∀n≥0,
and hence inequality (22) gives that
(24)∥xn+1-p∥2≤αn∥f(xn)-p∥2+(1-αn)∥xn-p∥2.

But we have that
(25)∥f(xn)-p∥2=[∥f(xn)-f(p)∥+∥f(p)-p∥]2≤[α∥xn-p∥+∥f(p)-p∥]2≤α2∥xn-p∥2+∥f(p)-p∥2+2α∥xn-p∥∥f(p)-p∥≤α(1+α)∥xn-p∥2+(1+α)∥f(p)-p∥2.

Substituting (25) into (24) we get that
(26)∥xn+1-p∥2≤(1-αn(1-α(1+α)))∥xn-p∥2+αn(1+α)∥f(p)-p∥2.

Therefore, by induction we get that
(27)∥xn+1-p∥2≤max⁡{∥x0-p∥2,1+α1-α(1+α)∥f(p)-p∥2},hhhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀n≥0,
which implies that {xn} and hence {yn} are bounded.

Let x*=PFf(x*). Then, from (17), Lemmas 1 and 7, and the methods used to get (22) we obtain that
(28)∥xn+1-x*∥2=∥αn(f(xn)-x*)+(1-αn)[anxn+bnTnyn+cnGxn-x*]∥2≤(1-αn)∥an(xn-x*)+bn(Tnyn-x*)hhhhhhhhhhh+cn(Gxn-x*)∥2+2αn〈f(xn)-x*,xn+1-x*〉≤(1-αn)bn∥Tnyn-x*∥2+(1-αn)an∥xn-x*∥2×(1-αn)cn∥Gxn-p∥2-(1-αn)anbn∥Tnyn-xn∥2-(1-αn)ancn∥Gxn-xn∥2+2αn〈f(xn)-x*,xn+1-x*〉,∥xn+1-x*∥2≤(1-αn)bn[∥yn-x*∥2+∥yn-Tnyn∥2]+(1-αn)(an+cn)∥xn-x*∥2-(1-αn)anbn∥Tnyn-xn∥2-(1-αn)ancn∥Gxn-xn∥2+2αn〈f(xn)-x*,xn+1-x*〉,≤(1-αn)bn[∥xn-x*∥2+λn2∥xn-Tnxn∥2]+(1-αn)bn[(1-λn)∥xn-Tnyn∥2hhhhhhhhhhhhhhh-λn(1-L2λn2-λn)hhhhhhhhhhhhhhh×∥xn-Tnxn∥2]+(1-αn)(an+cn)∥xn-x*∥2-(1-αn)anbn∥Tnyn-xn∥2-(1-αn)ancn∥Gxn-xn∥2+2αn〈f(xn)-x*,xn+1-x*〉
which implies that
(29)∥xn+1-x*∥2≤(1-αn)∥xn-x*∥2-(1-αn)bnλn[1-L2λn2-2λn]×∥xn-Tnxn∥2+(1-αn)bn(bn+cn-λn)∥xn-Tnyn∥2-(1-αn)andn∥Gxn-xn∥2+2αn〈f(xn)-x*,xn+1-x*〉(30)≤(1-αn)∥xn-x*∥2+2αn〈f(xn)-x*,xn+1-x*〉.

But
(31)〈f(xn)-x*,xn+1-x*〉=〈f(xn)-x*,xn-x*〉+〈f(xn)-x*,xn+1-xn〉≤〈f(xn)-f(x*),xn-x*〉+〈f(x*)-x*,xn-x*〉+∥xn+1-xn∥∥f(xn)-x*∥≤α∥xn-x*∥2+〈f(x*)-x*,xn-x*〉+∥xn+1-xn∥∥f(xn)-x*∥.

Thus, substituting (31) in (30) we obtain that
(32)∥xn+1-x*∥2≤(1-αn(1-2α))∥xn-x*∥2+2αn〈f(x*)-x*,xn-x*〉+2αn∥xn+1-xn∥·∥f(xn)-x*∥.

Next, we consider two cases.

Case 1. Suppose that there exists n0∈N such that {∥xn-x*∥} is decreasing for all n≥n0. Then, we get that {∥xn-x*∥} is convergent. Thus, from (29) and (23) we have that
(33)xn-Tnxn⟶0,Gxn-xn⟶0asn⟶∞.

Furthermore, from (17) and (33) we obtain that
(34)∥yn-xn∥=λn∥xn-Tnxn∥⟶0asn⟶∞,
and hence Lipschitz continuity of Tn, (34), and (33) implies that
(35)∥Tnyn-xn∥≤∥Tnyn-Tnxn∥+∥Tnxn-xn∥≤L∥yn-xn∥+∥Tnxn-xn∥⟶0hhhhhhhhhhhhhhhhhhhasn⟶∞.

Thus, from (33) and (35) we have that
(36)∥xn+1-xn∥=∥αn(f(xn)-xn)+(1-αn)×(anxn+bnTnyn+cnGxn)-xn∥≤αn∥f(xn)-xn∥+(1-αn)bn∥Tnyn-xn∥+(1-αn)cn∥Gxn-xn∥⟶0asn⟶∞.

Therefore, ∥xn+j-xn∥→0, as n→∞, for all j=1,2,…,M, and hence
(37)∥xn-Tn+jxn∥≤∥xn-xn+j∥+∥xn+j-Tn+jxn+j∥+L∥xn+j-xn∥⟶0,
as n→∞, for all j∈{1,2,…,M}.

Now, since {xn} is bounded subset of H, we can choose a subsequence {xnm} of {xn} such that xnm⇀x and limsup⁡n→∞〈f(x*)-x*,xn-x*〉=lim⁡m→∞〈f(x*)-x*,xnm-x*〉. Then, from (37) and Lemma 6 we have that x∈F(Tj), for each j=1,2,…,M. Hence, x∈∩j=1MF(Tj).

In addition, since G is nonexpansive, from Lemma 6 we get that x∈F(G) and hence by Lemmas 4 and 3 we obtain that x∈VI⁡(C,Aj), for each j∈{1,2,…,N}.

Therefore, by Lemma 5, we immediately obtain that
(38)limsup⁡n→∞〈f(x*)-x*,xn-x*〉=lim⁡m→∞〈f(x*)-x*,xnm-x*〉=〈f(x*)-x*,x-x*〉≤0.

Then, it follows from (32), (38), and Lemma 9 that ∥xn-x*∥→0 as n→∞. Consequently, xn→x*=PF(f(x*)).

Case 2. Suppose that there exists a subsequence {ni} of {n} such that
(39)∥xni-x*∥<∥xni+1-x*∥,
for all i∈N. Then, by Lemma 8, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, and
(40)∥xmk-x*∥≤∥xmk+1-x*∥,∥xk-x*∥≤∥xmk+1-x*∥,
for all k∈N. Now, from (29) and (23) we get that xmk-Tmkxmk→0 and Gxnk-xnk→0 as k→∞. Thus, following the method in Case 1, we obtain that xmk+1-xmk→0, xmk-Tjxmk→0, and
(41)limsup⁡k→∞〈f(x*)-x*,xmk-x*〉≤0.

Furthermore, from (32) and (40) we obtain that
(42)αmk(1-2α)∥xmk-x*∥2≤∥xmk-x*∥2-∥xmk+1-x*∥2+2αmk〈f(x*)-x*,xmk-x*〉+2αmk∥xmk+1-xmk∥∥f(xmk)-x*∥≤2αmk〈f(x*)-x*,xmk-x*〉+2αmk∥xmk+1-xmk∥∥f(xmk)-x*∥.

Now, using the fact that αmk>0 and (41) we get that
(43)∥xmk-x*∥2⟶0ask⟶∞,
and this together with (32) implies that ∥xmk+1-x*∥→0 as k→∞. Since ∥xk-x*∥≤∥xmk+1-x*∥ for all k∈N, we obtain that xk→x*. Hence, from the above two cases, we can conclude that {xn} converges strongly to a point x*=PFf(x*), which satisfies the variational inequality 〈(I-f)(x*),x-x*〉≥0, for all x∈F. The proof is complete.

If, in Theorem 10, we assume that f(x)=u∈C, a constant mapping, then we get the following corollary.

Corollary 11.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Tj:C→C, j=1,2,…,M, be Lipschitz pseudocontractive mappings with Lipschitz constants Lj, respectively. Let Aj:C→H, for j=1,2,…,N, be γj-inverse strongly accretive mappings. Assume that F=[∩j=1MF(Tj)]⋂[∩j=1NVI(C,Aj)] is nonempty. Let a sequence {xn} be generated from an arbitrary x0,u∈C by
(44)yn=(1-λn)xn+λnTnxn;xn+1=αnu+(1-αn)(anxn+bnTnyn+cnGxn),
where Tn=Tn(modM), G:=e0I+e1PC[I-γA1]+e2PC[I-γA2]+⋯+eNPC[I-γAr], for γ∈(0,2γ0), for γ0:=min⁡1≤j≤N{γj} with e0+e1+⋯+er=1, and bn+cn≤λn≤λ<1/(1+L2+1),∀n≥0, for L=max⁡{Lj:1≤j≤M}. Then, {xn} converges strongly to a unique point x*∈C satisfying x*=PF(u), which is the unique solution of the variational inequality 〈x*-u,x-x*〉≥0 for all x∈F.

If, in Theorem 10, we assume that N=1 and M=1, then we get the following corollary which is Theorem 3.1 of [21].

Corollary 12.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:C→C be Lipschitz pseudocontractive mappings with Lipschitz constant L and A:C→H an γ-inverse strongly accretive mapping. Let f:C→C be a contraction with constant α. Assume that F=F(T)⋂VI(C,A) is nonempty. Let a sequence {xn} be generated from an arbitrary x0∈C by
(45)yn=(1-λn)xn+λnTxn;xn+1=αnf(xn)+(1-αn)(anxn+bnTyn+cnPC[I-rA]xn),
where r∈(0,2γ) and bn+cn≤λn≤λ<1/(1+L2+1),∀n≥0. Then, {xn} converges strongly to a point x*∈F, which is the unique solution of the variational inequality 〈(I-f)(x*),x-x*〉≥0 for all x∈F.

If, in Theorem 10, we assume that Ti′s are strictly pseudocontractive mappins, then we get the following corollary.

Corollary 13.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→C, i=1,2,…M, be λi-strictly pseudocontractive mappings and let Ai:C→H, for i=1,2,…,N, be an γi-inverse strongly accretive mappings. Let f:C→C be a contraction with constant α. Assume that F=[∩i=1MF(Ti)]⋂[∩i=1NVI(C,Ai)] is nonempty. Let a sequence {xn} be generated from an arbitrary x0∈C by
(46)yn=(1-λn)xn+λnTnxn;xn+1=αnf(xn)+(1-αn)(anxn+bnTnyn+cnGxn),
where Tn=Tn(modM), G:=c0I+e1PC[I-γA1]+e2PC[I-γA2]+⋯+eNPC[I-γAr], for γ∈(0,2γ0), for γ0:=min⁡1≤i≤N{γi} with e0+e1+⋯+er=1, and bn+cn≤λn≤λ<1/(1+L2+1),∀n≥0, L=max⁡{(1+λi)/λi}. Then, {xn} converges strongly to a point x*∈F, which is the unique solution of the variational inequality 〈(I-f)(x*),x-x*〉≥0 for all x∈F.

If, in Theorem 10, we assume that Ti′s are nonexpansive mapping, then we get the following corollary.

Corollary 14.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→C, i=1,2,…,M, be nonexpansive mappings and let Ai:C→H, for i=1,2,…,N, be an γi-inverse strongly accretive mappings. Let f:C→C be a contraction with constant α. Assume that F=[∩i=1MF(Ti)]⋂‍[∩i=1NVI(C,Ai)] is nonempty. Let a sequence {xn} be generated from an arbitrary x0∈C by
(47)yn=(1-λn)xn+λnTnxn;xn+1=αnf(xn)+(1-αn)(anxn+bnTnyn+cnGxn),
where Tn=Tn(modM), G=c0I+e1PC[I-γA1]+e2PC[I-γA2]+⋯+eNPC[I-γAr], for γ∈(0,2γ0), for γ0:=min⁡1≤i≤N{γi} with e0+e1+⋯+er=1, an+bn+cn=1, and bn+cn≤λn≤λ<1/(2+1),∀n≥0. Then, {xn} converges strongly to point x*∈F, which is the unique solution of the variational inequality 〈(I-f)(x*),x-x*〉≥0 for all x∈F.

We note that the method of proof of Theorem 10 provides the following theorem which is a convergence theorem for a minimum norm point of common fixed points of a finite family of Lipschitz pseudocontractive mappings and common solutions of a finite family of variational inequality problems for accretive mappings.

Theorem 15.

Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let Tj:C→C, j=1,2,…,M, be Lipschitz pseudocontractive mappings with Lipschitz constants Lj, respectively. Let Aj:C→H, for j=1,2,…,N, be γj-inverse strongly accretive mappings. Assume that F=[∩j=1MF(Tj)]⋂‍[∩j=1NVI(C,Aj)] is nonempty. Let a sequence {xn} be generated from an arbitrary x0∈C by
(48)yn=(1-λn)xn+λnTnxn;xn+1=PC[(1-αn)(anxn+bnTnyn+cnGxn)],
where Tn=Tn(modM), G:=e0I+e1PC[I-γA1]+e2PC[I-γA2]+⋯+eNPC[I-γAr], for γ∈(0,2γ0), for γ0:=min⁡1≤j≤N{γj} with e0+e1+⋯+er=1, and bn+cn≤λn≤λ<1/(1+L2+1),∀n≥0, for L=max⁡{Lj:1≤j≤M}. Then, {xn} converges strongly to a unique minimum norm point x* of F(i.e., x*=PF(0)), which is the unique solution of the variational inequality 〈x*,x-x*〉≥0 for all x∈F.

4. Numerical Example

Now, we give an example of two Lipschitz pseudocontractive mappings and two γ-inverse strongly accretive mappings satisfying Theorem 10 and some numerical experiment result to explain the conclusion of the theorem as follows.

Clearly, for x,y∈C we have that
(50)〈(I-T1)x-(I-T1)y,x-y〉≥0,〈(I-T2)x-(I-T2)y,x-y〉≥0
which show that both mappings are pseudocontractive. Next, we show that T1 is Lipschitz with L=5. If x,y∈[-2,0], then
(51)|T1x-T1y|=|x+x2-y-y2|=|(x+y)+1||x-y|≤3|x-y|.

If x,y∈(0,2], then
(52)|T1x-T1y|=|x-y|.

If x∈[-2,0] and y∈(0,2], then
(53)|T1x-T1y|=|x+x2-y|=|x-y+x2|=|x-y+x2-y2+y2|=|x-y+x2-y2|+y2≤|x+y+1|·|x-y|+|y-x|2=(|x+y+1|+|x+y|)·|x-y|≤5|x-y|.

Thus, we get that T1 is Lipschitz pseudocontractive with L=5 and F(T1)=[0,2] which is not nonexpansive, since if we take x=-2 and y=-1.9, we have that |T1x-T2y|=0.29>0.1=|x-y|. Similarly, we can show that T2 is Lipschitz pseudocontractive with L=4 and F(T2)=[-2,1/2] which is not nonexpansive.

Furthermore, for C=[-2,2], let A1,A2:C→R be defined by
(54)A1x:={-(x-12)2,x∈[-2,12),0,x∈[12,2],A2x:={0,x∈[-2,23],3(x-23)2,x∈(23,2].

Then we first show that A1 is γ-inverse strongly accretive mapping with γ=1/5.

If x,y∈[-2,1/2), then
(55)〈A1x-A1y,x-y〉=〈-(x-12)2+(y-12)2,x-y〉=[(x-12)2-(y-12)2](y-x)=[(x-12)2-(y-12)2][(y-12)-(x-12)]=[(x-12)2-(y-12)2][(y-1/2)2-(x-1/2)2](y-1/2)+(x-1/2)=[(x-12)2-(y-12)2][(x-1/2)2-(y-1/2)2](1/2-x)+(1/2-y)≥15|(x-12)2-(y-12)2|2=15|A1x-A1y|2.

If x∈[-2,1/2) and y∈[1/2,2], we get that
(56)〈A1x-A1y,x-y〉=〈-(x-12)2,x-y〉=(x-12)2(y-x)=(x-12)2[(y-12)-(x-12)]≥(x-12)2(12-x)=(x-12)2(1/2-x)2(1/2-x)≥25|(x-12)2|2≥15|A1x-A1y|2.

If x,y∈[1/2,2], then we get that |A1x-A1y|=0 and hence
(57)〈A1x-A1y,x-y〉≥15|A1x-A1y|2.

Therefore, A1 is γ-inverse strongly accretive mapping with γ=1/5 and VI⁡(C,A1)=[1/2,2]. Similarly, we can show that A2 is γ-inverse strongly accretive mapping with γ=1/2 and VI⁡(C,A2)=[-2,2/3].

Note that we have F(T1)∩F(T2)∩VI⁡(C,A1)∩VI⁡(C,A2)={1/2}.

Thus, taking αn=1/(10n+100), λn=2/(n+100)+0.065, bn=cn=1/(n+100)+0.01, an=1-2/(n+100)-0.02, and f(x)=u∈C, we observe that conditions of Theorem 10 are satisfied and Scheme (17) provides the following Table 1 and Figures 1(a) and 1(b) for u=0.6 and u=0.8, respectively.

We observe that the data provides strong convergence of the sequence to the common point of fixed points of both pseudocontractive mappings and solutions of both variational inequality problems for γ-inverse strongly accretive mappings.

Table 1

u=0.6

x0=1

u=0.8

x0=-1

n

xn

n

xn

0

1.0000

0

−1.0000

500

0.6112

5000

0.0627

10,000

0.5137

10,000

0.4282

12,000

0.5121

15,000

0.4540

14,000

0.5110

20,000

0.4686

18,000

0.5093

25,000

0.4782

20,000

0.5087

35,000

0.4905

(a)(b)Remark 2.

Theorem 10 provides an iteration scheme which converges strongly to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems in Hilbert spaces.

Remark 3.

Theorem 10 improves Theorem 3.1 of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21] and Theorem 3.2 of Yao et al. [22] in the sense that our convergence is to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 167/130/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the referees for their valuable comments and suggestions, which improved the presentation of this paper.

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